Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
Annales Henri Lebesgue, Volume 5 (2022), pp. 275-301.

Metadata

Keywordsgrowth-fragmentation equation, self-similar fragmentation, measure solutions, long-time behavior, general relative entropy, Harris’s theorem, periodic semigroups

Abstract

We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptotic oscillations and that the convergence to this periodic behavior, in suitable spaces of weighted signed measures, occurs exponentially fast. It can be seen as a spectral gap result between the countable set of dominant eigenvalues and the rest of the spectrum, which is to our knowledge completely new. The two main difficulties in the proof are to define the projection onto the subspace of periodic (rescaled) solutions and to estimate the speed of convergence to this projection. The first one is addressed by using the generalized relative entropy structure of the dual equation, and the second is tackled by applying Harris’s ergodic theorem on sub-problems.


References

[AGG + 86] Arendt, Wolfgang; Grabosch, Annette; Greiner, Günther; Groh, Ulrich; Lotz, Heinrich P.; Moustakas, Ulrich; Nagel, R.; Neubrander, Frank; Schlotterbeck, Ulf One-parameter semigroups of positive operators, Lecture Notes in Mathematics, 1184, Springer, 1986 | DOI | MR | Zbl

[BA67] Bell, George I.; Anderson, Ernest C. Cell Growth and Division: I. A Mathematical Model with Applications to Cell Volume Distributions in Mammalian Suspension Cultures, Biophys. J., Volume 7 (1967) no. 4, pp. 329-351 | DOI

[BALZ18] van Brunt, Bruce; Almalki, A.; Lynch, T.; Zaidi, Ali A. On a cell division equation with a linear growth rate, ANZIAM J., Volume 59 (2018) no. 3, pp. 293-312 | DOI | MR | Zbl

[BCG13a] Balagué, Daniel; Cañizo, José A.; Gabriel, Pierre Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, Volume 6 (2013) no. 2, pp. 219-243 | DOI | MR | Zbl

[BCG20] Bansaye, Vincent; Cloez, Bertrand; Gabriel, Pierre Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions, Acta Appl. Math., Volume 166 (2020), pp. 29-72 | DOI | MR | Zbl

[BCG + 13b] Bardet, Jean-Baptiste; Christen, Alejandra; Guillin, Arnaud; Malrieu, Florent; Zitt, Pierre-André Total variation estimates for the TCP process, Electron. J. Probab., Volume 18 (2013), 10 | DOI | MR | Zbl

[BCGM19] Bansaye, Vincent; Cloez, Bertrand; Gabriel, Pierre; Marguet, Aline A non-conservative Harris ergodic theorem (2019) (https://arxiv.org/abs/1903.03946v1)

[BDJG19] Bernard, Étienne; Doumic Jauffret, Marie; Gabriel, Pierre Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, Kinet. Relat. Models, Volume 12 (2019) no. 3, pp. 551-571 | DOI | MR | Zbl

[Ber19] Bertoin, Jean On a Feynman–Kac approach to growth-fragmentation semigroups and their asymptotic behaviors, J. Funct. Anal., Volume 277 (2019) no. 11, 108270 | DOI | MR | Zbl

[BG20] Bernard, Étienne; Gabriel, Pierre Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, J. Evol. Equ., Volume 20 (2020) no. 2, pp. 375-401 | DOI | MR | Zbl

[BGP19] Broda, James; Grigo, Alexander; Petrov, Nikola P. Convergence rates for semistochastic processes, Discrete Contin. Dyn. Syst., Volume 24 (2019) no. 1, pp. 109-125 | DOI | MR | Zbl

[Bou18] Bouguet, Florian A Probabilistic Look at Conservative Growth-Fragmentation Equations, Séminaire de Probabilités XLIX (Donati-Martin, Catherine, ed.) (Lecture Notes in Mathematics), Volume 2215, Springer, 2018, pp. 57-74 | DOI | MR | Zbl

[BPR12] Banasiak, Jacek; Pichór, Katarzyna; Rudnicki, Ryszard Asynchronous exponential growth of a general structured population model, Acta Appl. Math., Volume 119 (2012), pp. 149-166 | DOI | MR | Zbl

[BW18] Bertoin, Jean; Watson, Alexander R. A probabilistic approach to spectral analysis of growth-fragmentation equations, J. Funct. Anal., Volume 274 (2018) no. 8, pp. 2163-2204 | DOI | MR | Zbl

[BW20] Bertoin, Jean; Watson, Alexander R. The strong Malthusian behavior of growth-fragmentation processes, Ann. Henri Lebesgue, Volume 3 (2020), pp. 795-823 | DOI | MR | Zbl

[Cav20] Cavalli, Benedetta On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes, Acta Appl. Math., Volume 166 (2020), pp. 161-186 | DOI | MR | Zbl

[CCC12] Carrillo de la Plata, José A.; Colombo, Rinaldo M.; Gwiazda, Piotr; Ulikowska, Agnieszka Structured populations, cell growth and measure valued balance laws, J. Differ. Equations, Volume 252 (2012) no. 4, pp. 3245-3277 | DOI | MR | Zbl

[CCC13] Cañizo, José A.; Carrillo de la Plata, José A.; Cuadrado, Sílvia Measure solutions for some models in population dynamics, Acta Appl. Math., Volume 123 (2013), pp. 141-156 | DOI | MR | Zbl

[CCF16] Campillo, Fabien; Champagnat, Nicolas; Fritsch, Coralie Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models, J. Math. Biol., Volume 73 (2016) no. 6-7, pp. 1781-1821 | DOI | MR | Zbl

[CCM10] Cáceres, María J.; Cañizo, José A.; Mischler, Stéphane Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces, Commun. Appl. Ind. Math., Volume 1 (2010) no. 2, pp. 299-308 | MR | Zbl

[CCM11] Cáceres, María J.; Cañizo, José A.; Mischler, Stéphane Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl., Volume 96 (2011) no. 4, pp. 334-362 | DOI | MR | Zbl

[CGY21] Cañizo, José A.; Gabriel, Pierre; Yoldaş, Havva Spectral gap for the growth-fragmentation equation via Harris’s Theorem, SIAM J. Math. Anal., Volume 53 (2021) no. 5, pp. 5185-5214 | DOI | MR | Zbl

[Clo17] Cloez, Bertrand Limit theorems for some branching measure-valued processes, Adv. Appl. Probab., Volume 49 (2017) no. 2, pp. 549-580 | DOI | MR | Zbl

[CMP10] Chafaï, Djalil; Malrieu, Florent; Paroux, Katy On the long time behavior of the TCP window size process, Stochastic Processes Appl., Volume 120 (2010) no. 8, pp. 1518-1534 | DOI | MR | Zbl

[CY19] Cañizo, José A.; Yoldaş, Havva Asymptotic behaviour of neuron population models structured by elapsed-time, Nonlinearity, Volume 32 (2019) no. 2, pp. 464-495 | DOI | MR | Zbl

[DBW12] Derfel, Gregory; van Brunt, Bruce; Wake, Graeme G. A cell growth model revisited, Funct. Differ. Equ., Volume 19 (2012) no. 1-2, pp. 75-85 | MR | Zbl

[DDJGW18] Debiec, Tomasz; Doumic Jauffret, Marie; Gwiazda, Piotr; Wiedemann, Emil Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation, SIAM J. Math. Anal., Volume 50 (2018) no. 6, pp. 5811-5824 | DOI | MR | Zbl

[DG20] Dumont, Grégory; Gabriel, Pierre The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states, Nonlinearity, Volume 33 (2020) no. 12, pp. 6381-6420 | DOI | MR | Zbl

[DHT84] Diekmann, Odo; Heijmans, Henk J. A. M.; Thieme, Horst R. On the stability of the cell size distribution, J. Math. Biol., Volume 19 (1984), pp. 227-248 | DOI | MR | Zbl

[DJB18] Doumic Jauffret, Marie; van Brunt, Bruce Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation, ESAIM, Proc. Surv., Volume 62 (2018), pp. 30-42 | DOI | MR | Zbl

[DJEM16] Doumic Jauffret, Marie; Escobedo Martínez, Miguel Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, Volume 9 (2016) no. 2, pp. 251-297 | DOI | MR | Zbl

[DJG10] Doumic Jauffret, Marie; Gabriel, Pierre Eigenelements of a General Aggregation-Fragmentation Model, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 5, pp. 757-783 | DOI | MR | Zbl

[DJHKR15] Doumic Jauffret, Marie; Hoffmann, Marc; Krell, Nathalie; Robert, Lydia Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, Volume 21 (2015) no. 3, pp. 1760-1799 | DOI | MR | Zbl

[EHM15] Evers, Joep H. M.; Hille, Sander C.; Muntean, Adrian Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differ. Equations, Volume 259 (2015) no. 3, pp. 1068-1097 | DOI | MR | Zbl

[EHM16] Evers, Joep H. M.; Hille, Sander C.; Muntean, Adrian Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM J. Math. Anal., Volume 48 (2016) no. 3, pp. 1929-1953 | DOI | MR | Zbl

[EMMRR05] Escobedo Martínez, Miguel; Mischler, Stéphane; Rodriguez Ricard, Mariano On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 1, pp. 99-125 | DOI | MR | Zbl

[Gab18] Gabriel, Pierre Measure Solutions To The Conservative Renewal Equation, ESAIM Proc. Surveys, Volume 62 (2018), pp. 68-78 | DOI | MR | Zbl

[GLMC10] Gwiazda, Piotr; Lorenz, Thomas; Marciniak-Czochra, Anna A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differ. Equations, Volume 248 (2010) no. 11, pp. 2703-2735 | DOI | MR | Zbl

[GM19] Gabriel, Pierre; Martin, Hugo Steady distribution of the incremental model for bacteria proliferation, Netw. Heterog. Media, Volume 14 (2019) no. 1, pp. 149-171 | DOI | MR | Zbl

[GN88] Greiner, Günther; Nagel, Rainer Growth of cell populations via one-parameter semigroups of positive operators, Mathematics applied to science (New Orleans, La., 1986), Academic Press Inc., 1988, pp. 79-105 | DOI | MR | Zbl

[GS14] Gabriel, Pierre; Salvarani, Francesco Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett., Volume 27 (2014), pp. 74-78 | DOI | MR | Zbl

[Hei84] Heijmans, Henk J. A. M. On the stable size distribution of populations reproducing by fission into two unequal parts, Math. Biosci., Volume 72 (1984) no. 1, pp. 19-50 | DOI | MR | Zbl

[Hei85] Heijmans, Henk J. A. M. An eigenvalue problem related to cell growth, J. Math. Anal. Appl., Volume 111 (1985) no. 1, pp. 253-280 | DOI | MR | Zbl

[HM11] Hairer, Martin; Mattingly, Jonathan C. Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on stochastic analysis, random fields and applications VI. Centro Stefano Franscini, Ascona, Italy, May 19–23, 2008 (Dalang, Robert C., ed.) (Progress in Probability), Volume 63, Springer, 2011, pp. 109-117 | DOI | MR | Zbl

[HW90] Hall, Adam J.; Wake, Graeme C. Functional-differential equations determining steady size distributions for populations of cells growing exponentially, J. Aust. Math. Soc., Volume 31 (1990) no. 4, pp. 434-453 | DOI | MR | Zbl

[LP09] Laurençot, Philippe; Perthame, Benoît Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., Volume 7 (2009) no. 2, pp. 503-510 | DOI | MR | Zbl

[Mal15] Malrieu, Florent Some simple but challenging Markov processes, Ann. Fac. Sci. Toulouse, Math., Volume 24 (2015) no. 4, pp. 857-883 | DOI | Numdam | MR | Zbl

[Mar19a] Marguet, Aline A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages, ESAIM, Probab. Stat., Volume 23 (2019), pp. 638-661 | DOI | MR

[Mar19b] Marguet, Aline Uniform sampling in a structured branching population, Bernoulli, Volume 25 (2019) no. 4A, pp. 2649-2695 | DOI | MR | Zbl

[Mic06] Michel, Philippe Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., Volume 16 (2006) no. 7, Suppl., pp. 1125-1153 | DOI | MR | Zbl

[MMP05] Michel, Philippe; Mischler, Stéphane; Perthame, Benoît General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl., Volume 84 (2005) no. 9, pp. 1235 -1260 | DOI | MR | Zbl

[Mon15] Monmarché, Pierre On 1 and entropic convergence for contractive PDMP, Electron. J. Probab., Volume 20 (2015), 128 | DOI | MR | Zbl

[MS16] Mischler, Stéphane; Scher, Justine Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 3, pp. 849-898 | DOI | MR | Zbl

[Per07] Perthame, Benoît Transport equations in biology, Frontiers in Mathematics, Birkhäuser, 2007 | DOI | MR | Zbl

[PPS14] Pakdaman, Khashayar; Perthame, Benoît; Salort, Delphine Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation, J. Math. Neurosci., Volume 4 (2014) no. 1, 14 | DOI | MR | Zbl

[PR05] Perthame, Benoît; Ryzhik, Lenya Exponential decay for the fragmentation or cell-division equation, J. Differ. Equations, Volume 210 (2005) no. 1, pp. 155-177 | DOI | MR | Zbl

[RP00] Rudnicki, Ryszard; Pichór, Katarzyna Markov semigroups and stability of the cell maturity distribution, J. Biol. Syst., Volume 8 (2000) no. 1, pp. 69-94 | DOI

[SS71] Sinko, James W.; Streifer, William A Model for Populations Reproducing by Fission, Ecology, Volume 52 (1971) no. 2, pp. 330-335 | DOI

[WG87] Webb, Glenn F.; Grabosch, Annette Asynchronous exponential growth in transition probability models of the cell cycle, SIAM J. Math. Anal., Volume 18 (1987) no. 4, pp. 897-908 | DOI | MR | Zbl

[ZBW15] Zaidi, Ali A.; Van Brunt, Bruce; Wake, Graeme C. Solutions to an advanced functional partial differential equation of the pantograph type, Proc. R. Soc. Lond., Ser. A, Volume 471 (2015) no. 2179, 20140947 | DOI | MR | Zbl